The Topology of Chaos - Department of Physics - Drexel University

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Abstract The Topology of Chaos Robert Gilmore Intro.-01 Intro.-02 Intro.-03 Exp’tal-01 Exp’tal-02 Exp’tal-03 Exp’tal-04 Exp’tal-05 Exp’tal-06 Exp’tal-07 Exp’tal-08 Embed-01 Embed-02 Embed-03 Data generated by a low-dimensional dynamical system operating in a chaotic regime can be analyzed using topological methods. The process is (almost) straightforward. On a scalar time series, the following steps are taken: 1 Unstable periodic orbits are identified; 2 An embedding is constructed; ⋆ ⋆ 3 The topological organization of these periodic orbits is determined; 4 Some orbits are used to identify an underlying branched manifold; 5 The branched manifold is used as a tool to predict the remaining topological invariants. This algorithm has its own built in rejection criterion.
Abstract - Key Point The Topology of Chaos Robert Gilmore Intro.-01 Intro.-02 Intro.-03 Exp’tal-01 Exp’tal-02 Exp’tal-03 Exp’tal-04 Exp’tal-05 Exp’tal-06 Exp’tal-07 Exp’tal-08 Embed-01 One soft spot in this analysis program is the embedding step. Different embeddings can yield different topological results. This makes the following question exciting: When you analyze embedded data: How much of what you learn is about the embedding and how much is about the underlying dynamics? This question has been answered by creating a representation theory of low dimensional strange attractors. It is now possible to totally disentangle the mechanism generating the underlying dynamics from topological structure induced by the embedding step. Embed-02 Embed-03
Page 1: The Topology of Chaos Robert Gilmor
Page 5 and 6: Experimental Schematic The Topology
Page 7 and 8: Ask the Masters The Topology of Cha
Page 9 and 10: Real Data The Topology of Chaos Rob
Page 11 and 12: Time Evolution The Topology of Chao
Page 13 and 14: Embeddings The Topology of Chaos Ro
Page 17 and 18: More Knotted Embeddings The Topolog
Page 19 and 20: Strange Attractor The Topology of C
Page 21 and 22: Skeletons The Topology of Chaos Rob
Page 23 and 24: Dynamics and Topology The Topology
Page 25 and 26: Evolution in Phase Space The Topolo
Page 27 and 28: Motion of Blobs in Phase Space The
Page 29 and 30: Fundamental Theorem The Topology of
Page 31 and 32: Fundamental Theorem The Topology of
Page 33 and 34: Birman-Williams Theorem The Topolog
Page 35 and 36: A Mechanism with Symmetry The Topol
Page 37 and 38: Aufbau Princip for Branched Manifol
Page 39 and 40: Dynamics and Topology The Topology
Page 41 and 42: Topological Analysis Program The To
Page 45 and 46: Determine Topological Invariants Th
Page 47 and 48: Determine Topological Invariants Th
Page 49 and 50: Perestroikas of Strange Attractors
Page 51 and 52: Perestroikas of Strange Attractors
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An Unexpected Benefit The Topology
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Representations The Topology of Cha
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Representation Labels The Topology
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Inequivalence in R 3 The Topology o
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Equivalences The Topology of Chaos
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Creating Isotopies The Topology of
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Mendeleev’s Table of the Chemical
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Stars The Topology of Chaos Robert
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An Experimental Study The Topology
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Basis Set of Orbits The Topology of
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Perestroikas of Branched Manifolds
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Torus and Genus The Topology of Cha
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Flows in Vector Fields The Topology
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Canonical Forms The Topology of Cha
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Labeling Bounding Tori The Topology
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Exponential Growth The Topology of
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Aufbau Princip for Bounding Tori Th
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Poincaré Section The Topology of C
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Represerntation Theory Redux The To
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Embeddings The Topology of Chaos Ro
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Extrinsic Embedding of Bounding Tor
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The Road Ahead The Topology of Chao
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Our Result The Topology of Chaos Ro
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Four Levels of Structure The Topolo
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Answered Questions The Topology of
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Unanswered Questions The Topology o
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The Topology of Chaos - Department of Physics - Drexel University

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